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f(f(x)) = exp(x) + x
#1
let f(x) be Coo.

f(f(x)) = exp(x) + x

what is f(x) ?

does the (only) fixpoint at - oo help ?

can f(x) be entire ?

is there a solution f(x) such that f(x) has no fixpoint apart from - oo ?

can f(x) be expressed in terms of tetration ?

is there a solution f(f(x)) = exp(x) + x with f(x) E Coo , f(x) mapping all reals to reals and f(x) having no fixpoint apart from - oo ?

does " a solution f(f(x)) = exp(x) + x with f(x) E Coo , f(x) mapping all reals to reals and f(x) having no fixpoint apart from - oo " imply that all derivatives are strictly positive reals ?

can f(x) be expressed in terms of pentation ?

does the substitution y = 1/x help ?
( trying to 'move the fixpoint' but problems occur e.g. exp(1/x) has a singularity at 0 ! ... on the other hand perhaps considering a certain angle towards the singularity it might work to give a real-analytic solution ? )

does the strategy lim n-> oo f(f(x)) = exp(x) + x + 1/n work ?

how about the carleman matrix method ?

this seems like a difficult problem ...

regards

tommy1729
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Messages In This Thread
f(f(x)) = exp(x) + x - by tommy1729 - 12/12/2009, 12:54 AM
RE: f(f(x)) = exp(x) + x - by bo198214 - 12/14/2009, 09:52 AM
RE: f(f(x)) = exp(x) + x - by tommy1729 - 12/14/2009, 09:47 PM



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